3. block. BV M. Bökstedt and H. Vosegaard, Notes on point-set topology, electronically available at

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1 Location Department of Mathematical Sciences,, G Main A T. Apostol, Mathematical Analysis, Addison-Wesley. BV M. Bökstedt and H. Vosegaard, Notes on point-set topology, electronically available at Banach xed point theorem and dierential equations Tue, 5/3., 9 12 Startup Discussion of topics related to the 2. block: your questions related to the lectures problems with exercises Presentations by the participants Existence and uniqueness of solutions to ordinary dierential equations. Reference J Arne Jensen: Fikspunktsætningen og eksistens af løsning til sædvanlige differentialligning. Noter til Mat After the presentation, comments and discussion, there will be time to work on the exercises that you did not get to during the last two sessions.

2 Dierentiability. Tue, 5.3., 12:30 15 After having emphasized continuity, we now get to dierentiability, a concept that cannot be dened on general topological spaces, but only on open subsets of Euclidean space (or spaces built out of those): From the basic year and also from math in high school, you already know about many dierentiable functions and you can certainly calculate the derivative or the partial derivatives of many functions. You have seen a denition of dierentiability of a function f : R R, and you have perhaps been working with the Jacobi matrix for functions f : R n R m. For a function like f(x) = sin(x), everything is clear, but what about { x sin(1/x) for x > 0 f(x) = 0 for x 0 For this function your denition of dierentiability from high school will tell you what to do, but what about functions of more than one variable? The function { xy 2 for (x, y) (0, 0) f(x, y) = x 2 +y 4 0 for (x, y) = (0, 0) has derivatives in all directions in particular it has all its partial derivatives, but it is not even continuous at (0, 0)! Dierentiable functions ought to be continuous, at least, and thus the right denition requires more than the existence of partial and directional derivatives. The aim of this part of the course is to give the right denition of dierentiability. We will emphasize an interpretation via approximations by linear maps or linear tangent spaces to the graph of the function. A Chapter and Exercises 1. In Apostol p. 345, the function f(x, y) = { xy 2 x 2 +y 4 for (x, y) (0, 0) 0 for (x, y) = (0, 0)

3 is studied. Prove, or convince yourself, that you understand Apostols proof, that this function has all directional derivatives, and that it is not continuous at (0, 0). A 12.4, 12.7, Complex dierentiability. Higher order derivatives. Taylor expansion. Wed, 6.3., 9 12 Startup Questions? Problems? concerning tuesday's lectures and/or exercise session. Complex dierentiable functions enjoy especially nice properties. In particular, the real and imaginary parts have to satisfy the Cauchy-Riemann dierential equations, and this condition is also sucient! Can one ensure dierentiability of a function of several variables by inspections of its partial derivatives? Yes, there is a sucient (but not necessary) condition, using the continuity of the partial derivatives. An important tool in the proof is the mean value theorem in one and in several variables, which is interesting for other purposes, as well. The partial derivatives of a function may be dierentiable and thus give rise to higher order derivatives. Under which conditions is the result independent of the order of dierentiation? And how can one use the 1st and higher order derivatives for approximation purposes? The answer is given by Taylor 1 's formula (best approximation by a multivariate polynomial of a given degree) and an estimation of the remainder term. Real functions on an open set U R n that are k-times dierentiable with continuous derivatives form a specic subspace C k (U, R) of the set of continuous functions on U. The sup-norm is replaced by a ner norm, that also takes care of the higher order derivatives and thus it requires more for a sequence to converge. Using the notion of dierentiability, we can dene smooth manifolds : A C k n-manifold is a topological n-dimensional manifold M such that there is an open cover M i I U i where the U i are coordinate neighborhoods, i.e., there are homeomorphisms φ i : U i V i R n, and for all i, j I, the change of coordinates map: φ i φ 1 j : φ j (U j U i ) φ i (U j U i ) is C k. 1 history/mathematicians/taylor.html

4 A Sections 5.16, , 12.6, 9.10 Exercises 1. Show: There is no C 2 -function f : R 3 R such that f(x, y, z) = (y 2 z, 2xyz, xy 2 + y). 2. A (interpret the result by a drawing); Determine the Taylor series of the function f(x, y) = x y x + y around (x 0, y 0 ) = (1, 1) without calculating derivatives! Use your result to determine D 1,1,2 (1, 1). Smooth manifolds. Wed, 6.3., 12:30 15 Smooth manifolds are a central concept for instance in the theory of dierential equations and dynamical systems, in physics and astronomy. We recap the denition, give some non-obious examples, and describe several results, some of them quite new: Although the denition of a dierentiable manifold does not require it to be a subset of some Euclidean space, there is always an embedding of an n-manifold in some Euclidean space (Whitney 2, 1936). Topological manifolds can have several dierent structures as dierentiable manifolds (Kervaire, Milnor 3, 1963). In 1982, Simon Donaldson 4 and Michael Freedman 5 proved that R 4 as a dierentiable manifold is fundamentally dierent from all other R n. This was a very surprising result - and gave them the mathematicians' Nobel Prize, the Fields Medal, in There are still open problems in this area, which have resisted attack for many years. We also describe some of those. 2 history/mathematicians/whitney.html 3 history/mathematicians/milnor.html 4 history/mathematicians/donaldson.html 5 history/mathematicians/freedman.html

5 1. L.Conlon. Dierentiable manifolds, A rst course, pp No Exercises Plan for the 4. block Date: 19./ The inverse function theorem. The implicit function theorem. Implicitely dened functions and their derivatives. The Lagrange and Kuhn Tucker methods for optimization. Course Evaluation. Discussion of demands for other math courses at the Ph.D.-level.